Is obvius that the mathematical concept of sequence of objects is formalized with the concept of indexed family (inside a set theory)

But my question is about who generalized the concept of Hyperoperation from the usual Goodstein 3-ary function (or the Knuth's uparrows) to some indexed families that satisfies some properties.

I'm tryng to continue the discussion started in the thread about the dustributive property of Bennet's operation family:

Who actually gave the first definition of when an indexed family of binary operations is an Hyperoperations family? I need the reference because I made some improvement in the definition while writing a paper about the Hyperoperations.

But my question is about who generalized the concept of Hyperoperation from the usual Goodstein 3-ary function (or the Knuth's uparrows) to some indexed families that satisfies some properties.

I'm tryng to continue the discussion started in the thread about the dustributive property of Bennet's operation family:

(05/27/2014, 08:22 PM)MphLee Wrote:(05/27/2014, 07:45 PM)andydude Wrote: @MphLee

Hyperoperations, in the general sense, are any sequence of binary operations that includes addition and multiplication. The commutative hyperoperations satisfy this property because and . That formula is the starting point, it is the definition of commutative hyperoperations. The fact that it contains addition and multiplication can be discussed and proved from the definition.

I'm even aware that the term Hyperoperations usually means (can be formalized as) an indexed family of binary operations whith addition, multiplication and exponentiation belonging to the image of the indexed family (the image of the family is defined to be the image of the set of indexes- set of ranks- via the indicization function).

This definition is the one I found on Wikipedia and is very smart even if it cuts the Commutative hyperoperations out of the game (Maybe we can make a weaker concept of Hyperoperations Family without the exponentiation requirement, I would call them Weak Hyperoperations Families)...

Anyways I'm very courious...I was not able to find references about this terminology and I did not even find who introduced this formal definition.

Who actually gave the first definition of when an indexed family of binary operations is an Hyperoperations family? I need the reference because I made some improvement in the definition while writing a paper about the Hyperoperations.

MSE MphLee

Mother Law \((\sigma+1)0=\sigma (\sigma+1)\)

S Law \(\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)\)